Problem: $f(x) = \dfrac{ \sqrt{ 12 - x } }{ \sqrt{ x - 7 } }$ What is the domain of the real-valued function $f(x)$ ?
Answer: First, we need to consider that $f(x)$ is undefined anywhere where either radical is undefined, so anywhere where either radicand (the expression under the radical symbol) is less than zero. The top radical is undefined when $12 - x < 0$ So the top radical is undefined when $x > 12$ , so we know $x \leq 12$ The bottom radical is undefined when $x - 7 < 0$ So the bottom radical is undefined when $x < 7$ , so we know $x \geq 7$ Next, we need to consider that $f(x)$ is undefined anywhere where the denominator, $\sqrt{ x - 7 }$ , is zero. So $\sqrt{ x - 7 } \neq 0$ , so $x - 7 \neq 0$ , so $x \neq 7$ So we have three restrictions: $x \leq 12$ $x \geq 7$ , and $x \neq 7$ Combining these three restrictions, we know that $7 < x \leq 12$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid 7< x \leq12\, \}$.